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INTBODUCTION.9d Depth of crest submergence in a drowned or submerged weir.Q Volume of discharge per unit of time.C, M, m, /i, a,f, etc., empirical coefficients.BASE FORMULAS.The following formulas have been adopted by the engineers named:oQMLH ZgH.Hamilton Smith (theoretical).Bazin, with no velocity of approach.Bazin, with velocity of approach. OLII S.Francis a (used here). CLH fL.Fteley arid Stearns.EQUIVALENT COEFFICIENTS.The relations between the several coefficients, so far as they can begiven here, are as follows:M is a direct measure of the relation of the actual to the theoretical weir discharge.8.02 /* 5.35 M.APPROXIMATE RELATIVE DISCHARGE OVER WEIRS.For a thin-edged weir, the coefficient C in the Francis formula is3.33 -o-.Let C' be the coefficient for any other weir, and a? therelative discharge as compared with the thin-edged weir, then.(1)or, as a percentage,« 100 x 30 C'.a The coefficient C of Francis includes all the constant or empirical factors appearing in theformula, which is thus thrown into the simplest form for computation.

10WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.This expression will be found convenient in comparing the effect Oudischarge of various modifications of the weir cross section. For abroad-crested weir with stable nappe, /i 2.64, see p. 121. The discharge over such a weir is thus seen to be 79.2 per cent of that for athin-edged weir by the Francis f onnula.REFERENCES.The following authorities are referred to by page wherever citedin the text:BAZIN, H., Recent experiments on flow of water over weirs. Translated by ArthurMarichal and J. C. Trautwine, jr. Proc. Engineers' Club Philadelphia, vol. 7,No. 5, January, 1890, pp. 259-310; vol. 9, No. 3, July, 1892, pp. 231-244; No. 4,October, 1892, pp. 287-319; vol. 10, No. 2, April, 1893, pp. 121-164.BAZIN, H., Experiences nouvelles sur l'e"coulement en deVersoir, 6me art., Annalesdes Ponts et Chausse'es, Memoires et Documents, 1898, 2me trimestre, pp. 121-264.This paper gives the results of experiments on weirs of irregular section.Bazin's earlier papers, published in Annales des Ponts et Chausse'es, 1888, 1890,1891, 1894, and 1896, giving results of experiments chiefly relating to thin-edgedweirs and velocity of approach, have been translated by Marichal and Trautwine.BELLASIS, E. S., Hydraulics.BOVEY, H. T., Hydraulics.FRANCIS, JAMES B., Lowell hydraulic experiments.FRIZELL, JAMES P., Water power.FTELEY, A., and STEARNS, F. P., Experiments on the flow of water, etc. Trans. Am.Soc. Civil Engineers, January, February, March, 1883, vol. 12, pp. 1-118.MERRIMAN, MANSFIELD, Hydraulics.RAFTER, GEORGE W., On the flow of water over dams. Trans. Am. Soc. Civil Engineers, vol. 44, pp. 220-398, including discussion.SMITH, HAMILTON, Hydraulics.THEORY OF WEIR MEASUREMENTS.DEVELOPMENT OF THE WEIR.The weir as applied to stream gaging is a special adaptation of milldam, to which the term weir, meaning a hindrance or obstruction, hasbeen applied from early times. The knowledge of a definite relationbetween the length and depth of overflow and the quantity also probably antedates considerably the scientific determination of the relationbetween these elements.In theory a weir or notch a is closely related to the orifice; in fact,an orifice becomes a notch when the water level falls below its upperboundary.THEOREM OF TORRICELLI.The theorem of Torricelli, enunciated in his De Motu GraviumNaturaliter Accelerate, 1643, states that the velocity of a fluid passingthrough an orifice in the side of a reservoir is the same as that whichwould be acquired by a heavy body falling freely through the verticala Commonly applied to a deep, narrow weir.

THEORY OF WEIR MEASUREMENTS.11height measured from the surface of the fluid in the reservoir to the,center of the orifice.This theorem forms the basis of hydrokinetics and renders the weirand orifice applicable to stream measurement. The truth of this proposition was confirmed by the experiments of Mariotte, published in1685. It can also be demonstrated from the laws of dynamics and theprinciples of energy. 0ELEMENTARY DEDUCTION OF THE WEIR FORMULA.In deducing a theoretical expression for flow over a weir it isassumed that each filament or horizontal lamina of the nappe is actuated by gravity acting through the head above it as if it were flowingthrough an independent orifice. In fig. 1 the head on the successiveorifices being H H H etc., and their respective areas A A%, A3,etc., the total discharge would beFIG. 1. Torricellian theorem applied to a weir.If the small orifices A be considered as successive increments of headIf, the weir formula may be derived by the summation of the quantitiesin parentheses. H comprises n elementary strips, the breadth of eachis. The heads on successive strips are,, etc., and the totalbecomeswhereT JT Al -}-A z, etc., for a rectangular weir.The sum of theseresHence the discharge isn* V n213 fThe above summation is more readily accomplished by calculus.a See Wood, Elementary Mechanics, p. 167, also p. 291.

12WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.APPLICATION OF THE PARABOLIC LAW OF VELOCITY TO WEIRS.The following elementary demonstration clearly illustrates the character of the weir:According to Torricelli's theorem (see fig. 1), the velocity (v) of aThis is thefilament at any depth (so) below surface will be v equation of a parabola having its axis OX vertical and its origin 0at water surface. .Replacing the series of jets by a weir with crest atX, the mean velocity of all the filaments will be the average ordinateof the parabola OPQ. The average ordinate is the area divided bythe height, but the area, of a parabola is two-thirds that of the circumscribed rectangle; hence the mean velocity of flow through the weiris two-thirds the velocity at the crest, i. e., two-thirds the velocitydue to the total head If on the crest. The discharge for unit lengthof crest is the head II, or area of opening per unit length, multipliedby the mean velocity. This quantity also represents the area of theparabolic velocity curve OPQX. The mean velocity of flow in thenappe occurs, theoretically, at two-thirds the depth on the crest.The modification of the theoretical discharge by velocity of approach,the surface curve, the vertical contraction at the crest, and the variousforms that the nappe may assume under different conditions of aeration, form of weir section, and head control the practical utility of theweir as a device for gaging streams.GENERAL FORMULA FOR WEIRS AND ORIFICES.aConsider first a rectangular opening in the side of a retaining vessel.The velocity of flow through an elementary layer whose area is Ldywill be from Torricelli's theorem:FIG. 2. Rectangular orifice.The discharge through the entire opening will be, per unit of time,neglecting contractions,(4)a For correlation of the weir and orifice see Merriman, Hydraulics, 8th edition, 1903, p. 144.

VERTICAL CONTRACTION.13This is a general equation for the flow through any weir or orifice,rectangular or otherwise, Q being expressed as a function of y. Inthe present instance L is constant. Integrating,-- //,*.(5)For a weir or notch, the upper edge will be at surface, H #, andcalling If2 IT in equation (5),(6)In the common formula for orifices, only the head on the center ofgravity of the opening is considered.Expressing Ha and H in terms of the depth H on the center ofgravity of the opening and the height of opening d, Merriman obtains,after substituting these values in and expanding equation (5) by thebinomial theorem, the equivalent formula,.(7)The sum of the infinite series in brackets expresses the error of theordinary formula for orifices as given by the remainder of the equation. This error varies from 1.1 per cent when H d to 0.1 per centwhen H 3d.VERTICAL CONTRACTION.Practical weir formulas differ from the theoretical formula (6) inthat velocity of approach must be considered and the discharge mustbe modified by a contraction coefficient to allow for diminished section of the nappe as it passes over the crest lip. Velocity of approachis considered on pages 14 to 20. Experiments to determine the weircoefficient occupy most of the remainder of the paper. The nature ofthe contraction coefficient is here described.Vertical contraction expresses the relation of the thickness of nappe,s, in the plane of the weir crest, to the depth on the crest, //. If theratio s\H were unity, the discharge would conform closely with theexpressionThe usual coefficient in the weir formula expresses nearly the ratioslH.The vertical contraction comprises two factors, the surface curve ordepression of the surface of the nappe and the contraction of theunder surface of the nappe at the crest edge. The latter factor in

14WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.particular will vary with form of the weir cross section, and in general variation in the vertical contraction is the principal source ofvariation in the discharge coefficient for various forms of weirs.The usual base weir formula, Q 2i3 LHwgH, is elsewhere givenfor an orifice in which the upper edge is a free surface. If insteadthe depth on the upper edge of the orifice is d, the surface contraction,there results the formula(8)This is considered as the true weir formula by Merriman. a In thisformula only the crest-lip contraction modifies the discharge, necessitating the introduction of the coefficient. The practical difficulties ofmeasuring d prevent the use of this as #, working formula.Similarly a formula may be derived in which only the effectivecross section s is considered, but even this will require some correctionof the velocity. Such formulas are complicated by the variation of 8and d with velocity of approach. 6 Hence, practical considerationsincluded, it has commonly been preferred to adopt the convenient2base formula for weirs, Q g MLH gH, or an equivalent, and throwall the burden of corrections for contraction into the coefficient M.VELOCITY OF APPROACH.THEORETICAL FORMULAS.Before considering the various practical weir formulas in use somegeneral considerations regarding velocity of approach and its effect onthe head and discharge may be presented.In the general formula (4) for the efflux of water when the waterapproaches the orifice or notch with a velocity -y, then with free discharge, writing D-\- h in place of H, for a rectangular orifice, we have(9)DI and Z 2 being the measured depth on upper and lower edges of theorifice, and h , the velocity head. To assume that D-\- h equals H is to assume that the water level isa Hydraulics, 8th edition, p. 161.&See Trautwine and Marichal's translation of Bazin's Experiments, pp, 231-307, where may also befound other data, including a resum6 of M. Boussinesq's elaborate studies of the vertical contractionof the nappe, which appeared in Comptes Rendus de 1' Academic des Sciences for October 24, 1887,

VELOCITY OF APPROACH.15increased by the amount A, or, as is often stated, that jETis "measuredto the surface of still water." This is not strictty correct, however, because of friction and unequal velocities, which tend to makeH D h, as explained below.For a weir, Dj equals zero; integrating, 2Since Q L SgH, we have.(90)This is the velocity correction formula used by James B. Francis. aSince h appears in both the superior and inferior limits of integration, it is evident that A increases the velocity only, and not the section of discharge. The criticism is sometimes made that Francis'sequation has the form of an increase of the height of the section ofdischarge as well as the velocity.The second general method of correcting for velocity of approachconsists of adding directly to the measured head some function of thevelocity head, makingin the formulaor*).95This is the method employed by Boileau, Fteley and Stearns, andBazin. No attempt is made to follow theory, but an empirical correction is applied, affecting both the velocity and area of section.By either method v must be determined by successive approximations unless it has been directly measured.Boileau and Bazin modify (9J) so as to include the area of section ofchannel of approach, and since the velocity of approach equals QlA,a separate determination of v is unnecessary. Bazin also combinesthe factor for velocity of approach with the weir coefficient.The various modifications of the velocity correction formulas aregiven in conjunction with the weir formulas of the several experimenters.o Boyey gives similar proof of this formula for the additional cases of (1) an orifice with free discharge, (2) a submerged orifice, (3) a partially submerged orifice or drowned weir, thus establishingits generality.

16WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.DISTRIBUTION OF VELOCITY IN CHANNEL OF APPROACH.The discharge over a weir takes place by virtue of the potentialenergy of the layer of water lying above the level of the weir crest,which is rendered kinetic by the act of falling over the weir. If thewater approaches the weir with an initial velocity, it is evident thatsome part of the concurrent energy will facilitate the discharge.The theoretical correction formulas may not truly represent theeffect of velocity of approach for various reasons:1. The fall in the leading channel adjacent to the measuring sectionis the source of the velo

.H" The head corrected for the effect of velocity of approach, or the observed head where there is no velocity of approach. As will be explained, D is applied in formulas like Bazin's, in which the correction for velocity of approach is included